FE modelling of Wood Structure Petr Koňas 3.června 2008, Dolní Maxov Ing. Petr Koňas, Ph.D.

FE modelling of Wood StructurePetr Koňas

3.června 2008, Dolní MaxovIng. Petr Koňas, Ph.D.

Programy a algoritmy numerické matematiky 14

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Guidelines• Why to model wood structure?• Problems of geometry

– Disadvantages of homogenized models.– Problems in modelling of biol.structures.– Problem of anisotropy.– How deep? Low limits in modelling.– Problem of variability.– Probable structure – probable results.– Accuracy– How far? Up limits in modelling.– What a time?– Advantages and disadvantages of L-systems

• Coupled physical task • Weak solution with mixed multiscaled finite elements


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Why model wood structure?

- Homogenized models fails- Common models may be too complicated for accurate solution- Although the physical model exists on lower scale, it can be difficult to find

appropriate solution on higher scale(s)- Microscopic and lower scaled structures became more important than before mainly for nanotechnology and material science- Evaluation of material model “ab initio” may be more efficient than expensive experiments- Wood and several other parts are structures with remodelling character and

its development in time (growing) is closely connected to micro and meso-scales.

- Modelling on lower scales can help us to understand more the substance and reasons of final behaviour of wood.


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Disadvantages of homogenized models- Reduce information with local character, even though it can influence higher

scales- Wood is anisotropic material with high variability. It is difficult to determine

appropriated reference region.- Material properties embody variability according to scales. Accuracy of solution

on some scale level can be very poor in extension on lower or higher scales. Also accuracy on the same scale can be far off constant character.

- When some reference region is chosen, the problem of material properties is still not solved.


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• Generally insufficient amount of input parameters and verifying experiments.• How can we simply describe geometry of biol. structures? • How to describe variability of material properties (according to structure, scale, environment)?• How to describe relationship of material and geometry properties on scale? • How to determine influence of properties from specific scale on behaviour on higher scales? (vegetation × abiotic factors, stand × fluid flow, tree × bush, crown/roots × stem, heartwood × sap, early × late wood, tracheids × parenchyma cells, lumen × cell layers, lignin × cellulose, covalent × ion coupling, quantum effects)

• How to involve “individuality” and “stereotype”? (selfsimilar/fractal structure × individual occurrence/mutation)• How to involve genetic predetermination and phenotypic resignation?

Problems in modelling of biol.structures


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Problem of anisotropyAnisotropic homogenized models of wood. Simple geometry allows declaring of complex constitutive relationships

FE model of anatomy structure of early/late wood. Detail geometry allows to determine influence of small structures (pits) on global behaviour. Complex physical model can be used.

Early & late wood FE models include variability on wider range, but complexity of physics has to be lower.

FE model of homogenized micro structure (superelements) tends to be sufficient estimation of material on macro scale. (time-consuming)


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How deep?

Early and late type of tracheid

Early and late wood transition

Cell wall with layered elements

Orientation of fibrils in individual layers


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Problem of variabilityProbabilistic structure

How to determine structure of wood for arbitrary position?If statistical behaviour of morphological parameters can be determined, the cross-scale relationships (transition functions) can be evaluated.


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Near the stem pith Far off the stem pith

Shear RL Shear RT Shear TL

etc.

Probable structure – probable results

Tensionx

Compression


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Accuracy?

Accuracy and large amount of material characteristics provide “accurate” results on statistical sense. “Accuracy” can be also in 30% of variation coefficient. This problem origins from assumption, that geometry has stochastic character (this assumption coincides with the similar statement that material properties has stochastic character), but geometry is consequence of developing stage (growing) and material in every position in stem is not independent, but it is related to position, environment, conditions, etc.Probabilistic FE model are unsuitable for homogenization and solution on higher scales.

Solution of such problem can be evolutionary models.


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How far?

A=aaSaaSaaSaaSaaSaaSaaSaaSaaSaaBS='(.9)!(.9)a=tF[&'(.8)!B|z]>(137)[z&'(.7)!B|z]>(137)B=tF[-'(.8)!(.9)$C|z]'(.9)!(.9)CC=tF[+'(.8)!(.9)$B|z]'(.9)!(.9)B

A=F[&'(.7)!B]>(137)[&'(.6)!B]>(137)'(.9)!(.9)AB=F[-'(.7)!(.9)$C]'(.9)!(.9)CC=F[+'(.7)!(.9)$B]'(.9)!(.9)B

c(12)FFFFFFFFFF>(1)&(1)AA=!(.75)t(.9)FB>(94)B>(132)BB=[&"t(.9)!(.75)F[|z]$A[|z]]

c(12)AA=aaSaaSAS='(.8)!(.9)a=tF[&'(.8)!B]>(137)[z&'(.7)!B]>(137)B=tF[-'(.8)!(.9)$C]'!(.9)CC=tF[+'(.8)!(.9)$B]'!(.9)B

c(12)FFFFFAA=!(.8)tFB>(94)C>(132)DB=[&'t(.5)!(.9)F$A|z]C=[&'t(.4)!(.8)F$A|z]D=[&'t(.3)!(.7)F$A|z]

Orthotropic material properties, gravitational acceleration, non axis force, base of stem is fixed, 840-1500 geometric components, 100 000-1 200 000 finite elements, 5 mil.DOFs.

Lindenmayer grammar allows to create iterated structures with selfsimilar/fractal character as on micro scale as on macro scale. With FE translator very complex geoms can be numerically solved.


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What a time?

A =

!(.

8)tF

B>

(94)

C>

(132

)DB

= [

&'t(

.5)!

(.9)

F$A

L|z

L]

C =

[&

't(.4

)!(.

8)F$

AL

|zL

]D

= [

&'t(

.3)!

(.7)

F$A

L|z

L]

L =

[~c

(8){

+(3

0)f(

.5)-

(120

)f(.

5)-(

120)

f(.5

)}]

A =

aaSaaS

aaSaaS

aaSaaS

aaSaaS

aaSaaB

S =

'(.9)!(.9)a =

tF[&

'(.8)!LB

L|zL

]>(137)[z&

'(.7)!LB

L|zL

]>(137)

B =

tFL

[-'(.8)!(.9)$LC

L|zL

]'(.9)!(.9)CC

= tF

L[+

'(.8)!(.9)$LB

L|zL

]'(.9)!(.9)B

L =

[~c(8){+(30)f(.4)-(120)f(.4)-(120)f(.4)}]

L-systems

Also evolutionary models can be analyzed by Lindenmayer systems


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Advantages and disadvantages of L-systems

Advantages- L-systems seems to be very realistic (according to geometry and description of material

properties distribution)- Models derived by iterated functions include less unknowns and offers tool for evaluation

of variables which depends on geometry (permeability, variation of properties)- Model assembling can be less data-intensive - Predictive abilities can be higher in comparison with homogenized models

Disadvantages- L-system based models translated finite element mesh are still very huge- Formed models are usually very complicated- Iterated function for geometry of structure has to be revealed.- Solution is very difficult for coupled physical problems- It is multispecialty, money and time consuming problem


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Coupled physical task

One of the most difficult problem in Wood Science is modelling of microwave wood drying process including micro-effects on small scales.

2'' E effabsq

qabs is density of energy, is angular velocity (s-1), ‘’eff is effective relative loss factor, E is electric field (V.m-1)

0

B

D

DJH

BE

e

t

t

B is the magnetic flux density, D is electric flux density, H is magnetic field intensity, J is current density, re is electric charge density. Due to anisotropy of wood we can itemize these variables to EJHBED , , , where e is permittivity, is permeability and is electricconductivity of material.


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Coupled physical taskWhen only conduction due to the microwave heating source is considered:

absqTt

TC

k

When also convective source plays important role:

TTkqTtT

C exthabs T

k

Temperature is not only one of important fields, which is changing during drying process

ppTpwt

p

wwTpwt

w

TTqTpwt

TC

ext

ext

extabs

p

w

T

hpTpppw

hwTwpww

hTTTpTw

kkkk

kkkk

kkkk


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Coupled physical taskRapid change of moisture, temperature are reasons for large time-depended stress-strain effect.

t

t

vel

celc

εDε

σ

elc is immediate elastic strain (structural), vel

c is viscous-elastic part of strain (structural), F() is function of memory effect, D is matrix of

elasticity, is relaxation time

Both strains are composed from pure mechanical, thermal and moisture components

velT

velw

velvelc

elT

elw

elelc

εεεε

εεεε

In our case the problem was simplified for visco-elastic strains

velTw,

velc εε

or for constant memory effectt

elc

velTw,

Tw,

ελDεσ


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Elastic components has usual linear character

βε

αε

elT

elw

extext

elw

elwel

welw

elw

TTTT

HLwk

HLkk

HLw

kHLw

k

654321

654321

|||||

for 0

for w 1,1|||||1

Also elastic mechanical properties for pure mechanical behaviour should include linear influence of moisture and temperature

extbextbrw

extbextbrw

TTkwwkGG

TTkwwkEE

Tj

wjjj

Ti

wiii

HLwk

HLwkk

wi

wi

wi

b

b

b for 0

for 0


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Coupled physical taskThan modified matrix of elasticity can be formed

6

5

4

3,,2,,,1,,,

3,,,2,,1,,,

3,,,2,,,1,,

00000

00000

00000

0001

0001

0001

X

X

Xk

X

k

X

k

Xk

X

k

X

k

Xk

X

k

X

k

X

D

yxxy

D

yxxzyz

D

yzxyxz

D

zxxyzy

D

zxxz

D

zyxzxy

D

zyyxzx

D

zxyzyx

D

zyyz

XD

1,,,,,,,,,,,, zxxzzyyxxzzxyzxyyxxyzyyzDk 654321 ,,,,, XXXXXXX

In this declaration we used kD as constant with the following meaning

and vector for simple substitution during separation.

Matrix of elasticity can be simply defined:

Tbwb

KXXKXXEGXX DDDD

extext TTww

withal

TTTTTTwwwwww bbbbbbbbbbbb

xzyzxyzyx

kkkkkkkkkkkk

GGGEEE

654321654321,,,,, respective ,,,,,

,,,,,

Tw bb

rr

KK

GEEG

Final stress-strain relation on modified matrixes of elasticity

t

TTHL

wkTTww ext

elwextext

velTw,

Tw,el

KKEG

ελβαεDDDσ

Tbwb

1


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Coupled physical taskPrevious equation has to be disassembled into unknown displacement by the common relationships.

ij

ij Fxt

2

2u

velvelvelvel wvu

z

u

x

w

y

w

z

v

x

v

y

u

z

w

y

v

x

u

z

u

x

w

y

w

z

v

x

v

y

u

z

w

y

v

x

u

,,

2

1

2

1

2

1

2

1

2

1

2

1

111111111

u

ε

ε

velTw,

el

u is vector of displacements wvu ,,u , Fi are components of volume forces


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The final relationship for stress-strain components according to unknown variable displacement u respective uvel can be formed in this grouped form.

FCCCCCC

ucuccc

u

wTTTww

velλKKEG

22

Tw,Tbwb

wTTTww

tTTww

t extext

22

2

2

βαDDDCαDβDC

βDCβαDβDDC

αDCβαDαDDC

TbwbTbwb

Tb2

Tbwb

wb2

wbTb

KKEGKKwT

KTKKEGT

KwKKEGw

extelwextext

elw

extelwext

elw

extextel

wext

elw

TkTwHL

k

Tkw

HL

kT

HL

wkT

HL

k

;

;2

;1

333231

5521

5521

6621

6621

5521

5521

232221

4421

4421

6621

6621

4421

4421

131211

000000

0000000

0000000

0000000

000000

0000000

0000000

0000000

000000

XXX

XX

XX

XX

XXX

XX

XX

XX

XXX

DDD

DD

DD

DD

DDD

DD

DD

DD

DDD

Xc

TTbbwb

wb KXXKKXXKEGXXEG cccccc

,,

3541

641

521

541

621

641

541

521

541

2441

421

441

641

621

441

421

641

441

1

000000

0000000

0000000

0000000

000000

0000000

0000000

0000000

000000

Tw,c


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ppTpwt

p

wwTpwt

w

TTqTpwt

TC

ext

ext

extabs

p

w

T

hpTpppw

hwTwpww

hTTTpTw

kkkk

kkkk

kkkk

2'' E effabsq

Described model is valid for diffusive transport of moisture and temperature. It is not appropriate (due physical nature of phenomenon) for free water movement. This transport is allocated into intercellular spaces and cell lumen. Description of this process can be done with Navier-Stokes equation. Finally the following set of PDE’s has to be solve:

flptzyx

ν

νF

ν 22

2

1

FCCCCCC

ucuccc

u

wTTTww

velλKKEG

22

Tw,Tbwb

wTTTww

tTTww

t extext

22

2

2


strana 22/33

0,2

,,,

22

2

2

ξCCCCCCF

ξu

cξucccξu

wTTTww

velλKKEG

22

Tw,Tbwb

wTTTww

tTTww

tF extextu

The weak form of thermal-moisture displacements can be written as follows:

0H , for all and meaning of as scalar product on Hilbert space.

1

11m

VVVV m 1

1

11 ,,, 21 ijV j

j 1for

VVVVVV m 11

1

111 ,,321

Let us assume the region is partitioned by linear mesh on very fine scale also we will assume that region is not of small regions are covered by mesh on this scale (subgrids).

, where are Raviart-Thomas (RT) spaces.

fully partitioned by this fine mesh. Only

Functional is than defined on vector subspaces Subspaces may not fill the full space V. It means that

1

1

1

11

1

1

1

1

11

22

1

2

1

2

11

11

1

1

1

1 ,2,1,2,2,1,1,2,1, ,,,,,,,, mnVVVnVVVnVVV VVVmmmm

11

11

1

1

1

1

1

22

1

2

1

2

1

11

1

1

1

1

1,2,1,,2,1,,2,1, ,,,,,,, V

mmmm nVVVnVVVnVVV 1V

Withal we declare mentioned vector subspaces with bases

Complete basis on vector space

i ,,,

32

i 21 immm ,,, 32

VVVVVVVVVVVV iiimmm

2

3

2

332

2

22 ,,,,,,,,,,,, 212121

Similarly let us to partition by next linear meshes for different scales where again

regions cover some parts of on specific scale. Consequently similar vector subspaces can be distinguished

with the same requirements:

.,,

,

,,,

,,,

2

23

2

33

12

2

22

21

21

21

VVVVV

VVVVV

VVVVV

iiiim

m

m

Weak solution


strana 23/33

1

1 from 1 V

1

1 from 2 V

2

2 from 1 V 2

2 from 2 V

3

3 from 1 V


strana 24/33

Weak solution

VfA uu :

uf,uu, 2 AuF

All unknowns can be decomposed to individual scales e.g.: oi someon 21 uuuu

Decomposition of unknowns to individual scales affects solution in sense of finite elements and minimisation of functional (1) does not provide common appearance of Ritz system

Let us consider PDE with differential operator A and follow common steps in solution of this task for multi-scale problem.

Functional which will be minimized has standard form:

Decomposed unknown will be substituted into first part of functional: AuiiL uuuuuu 2121 ,

A

AA

AAA

u

ii

i

i

uu

uuuu

uuuuuu

L

,

,2,

,2,2,222

12111

ji

jjj

s

kkkb

1

~ u

1

j

jjj

s

kkka

1

u

It can be expanded due to rules of scalar product in the following manner.

As usual the functional is minimized by the function:

For first step we will approximate functional in subgrid on scale

Finally unknown function can be by this function:


strana 25/33

ju ~

k

k

j

j

k

k

j

j

k

k

jk

k

jkjkj

k

k

jk

k

jkjkjkjkj

kj

ssss

ss

ss

bb

bbbb

bbbbbb

uu

,

,2,2

,2,2,

~,~222222

1121211111

kj for

2

22

2

222

112121

2

111

,

,2,2

,2,2,

~,~

j

j

j

j

j

j

j

j

jj

j

jjjj

j

j

jj

j

jjjjjjjj

jj

sss

ss

ss

b

bbb

bbbbb

uu

iis

iis

iiss

i

iiis

iis

iiss ababababs

u

abababab

u

b

L

b

L

11

11

11

11

1111

11

11

11

11

1

,,,,1

,,

kj

a

a

a

buu

buu

buu

j

j

j

j

k

k

j

j

k

k

jk

k

j

kjkj

kj

k

k

kj

k

kj

k

kj

sAssAsAs

AA

A

s

for

,,2,2

0,,2

00,

~,~

~,~

~,~

2

1

21

2221

11

2

1

Evaluation Lu for minimizing function can be done on these relationships:

Requirement on minimisation of quadratic functional Fu allows evaluating a minimum of function. Thus partial differentiation according to all coefficients on all scales should be done.This task can be easily achieved by next relations.

jkjk

k

k

kjLA

sbbuu

aR

1

~,~

LA

kjR is modified lower triangular

matrix of Ritz system.

kj

a

a

a

b

uu

b

uu

b

uu

k

k

k

k

k

k

j

j

k

k

jkj

k

k

jkjkj

j

j

kj

j

kj

j

kj

sAss

AsA

AsAA

s

for

,00

,2,0

,2,2,

~,~

~,~

~,~

2

1

222

12111

2

1

kkjj

j

j

kjUA

sbb

uu

aR

1

~,~

UA

kjR is modified upper triangular

matrix of Ritz system.

j

j

j

j

j

j

j

j

j

j

jj

j

j

j

j

jjjjj

j

j

jjjjj

j

j

jj

j

jj

j

jj

sAssAsjAs

AsAA

AsAA

s

a

a

a

b

uu

b

uu

b

uu

2

1

1

22221

12111

2

1

,,,

,,,

,,,

2

~,~

~,~

~,~

jjjj

j

j

jj

A

sbb

uu

aR

1

~,~

kjA

R is well known matrix of Ritz system


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Weak solutionApplying of mentioned rules on left part of functional leads to:

iii

i

i

ii

ii

ii

ii

A

LA

LA

s

u

UA

UAA

LA

LA

s

u

UA

UAA

LA

s

u

UA

UAA

s

u

bb

L

bb

L

bbL

bb

L

aR

aR

aR

aRaRaR

aR

aR

aRaRaR

aR

aRaRaR

2

2

22

2

2

22

2

22

22

11

2

3443333

232

131

3

2

3

2332222

121

2

2

2

12211111

1

1

1

1

1

1

This complex system can be rewritten in more readable form:

1

112

1

22j

k

LA

i

jk

UAA

s

u

u

u

A kjkkkjjjj

j

j

j

j

b

L

b

Lb

L

S

aRaRaR


strana 27/33

j

j

j

j

kjkkkjjjj

j

j

j

j

s

j

k

LA

i

jk

UAA

s

u

u

u

A

f

f

f

b

L

b

Lb

L

S

,

,

,

22 2

1

1

112

1

aRaRaR

0,,0

1111

11

11

1111

11

11

11

11

1

,,,,1

iis

iis

iiss

i

iiis

iis

iiss ababababs

u

abababab

u

b

F

b

F

ja

j

j

j

j

sc

c

c

CBA

f

f

f

SSS

,

,

,

2 2

1

When the full functional is minimized by:

The solution of the initial problem can be reached by enumeration of

By analogy, the solution of coupled problem with applying of SA derivation can be rewritten.

for differential operator

0,2

,,,

22

2

2

ξCCCCCCF

ξu

cξucccξu

wTTTww

velλKKEG

22

Tw,Tbwb

wTTTww

tTTww

tF extextu

tC

TTwwBt

A extext

Tw,

Tbwb

λ

KKEG

c

ccc ,,2

2

CCCCCCF wTTTww 22 wTTTwwf c22

For differential operators

and function


strana 28/33

Weak solutionSolution is realized in i consequent steps of solution. In first step the previous equation is formed, whereas results of higher scales are unknown (in Ritz or modified Ritz system). Solution on higher scales in individual nodes can be expressed by mapping of

1a

From this step we obtain definitions in some nodes on higher scale(s) which bounds region of element on this solved scale. In the following we calculate the same eq., but on the following higher scale withal some nodes on this scale were strictly derived from previous step. This idea is repeated until the highest scale is reached. Advantage of this type of solution is also that you do not need enumerate results on lower scales, but you can enumerate only results on last scale whereas results on this scale is derived from the low and lower scales.

or other appropriate lower scales.


strana 29/33

Coupled microwave drying of woodnon-scaled problem

EMAG task Heating task


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Coupled microwave drying of woodnon-scaled problem


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Computational sourcesNumerical simulations are very source-consuming processes. Usable and appropriate models consists of more than 3mil. DOF’s. From this reason Dep. of Wood Science on Mendel University built together with Dep. of Theoretical and experimental electrotechnics on Technical University in Brno cluster for high performance computing. We also participate on national grid project (METACENTRUM) for extensive distributed tasks. Finally the EU grid EGEE for scientific computations became big source for our computing.

16 CPU AMD64(Dp.WS Mendel Univ. & Dp.TEE

Technical Univ.)

500 CPUMETACENTRUM (FI MU)

Dp.WS Mendel Univ.

2500 CPU(EGEE Grid EU)


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Acknowledgment

The Research project GP106/06/P363 Homogenization of material properties of wood for tasks from mechanics and thermodynamics (Czech Science Foundation) and Institutional research plan MSM6215648902 - Forest and Wood: the support of functionally integrated forest management and use of wood as a renewable raw material (2005-2010, Ministry of Education, Youth and Sport, Czech Republic) supported this work.


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Thank you, for your attention...


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J. da Cimrman „Teorie externismu“

...okolí není tam, kde si myslíme, že je ale je přesně tam, kde si myslíme, že není…

YYXXX VVVWWNE Esence & Esence K. Gödel’s statement of God’s existence

Everybody has its own truth on some „scale“

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FE modelling of Wood Structure Petr Koňas 3.června 2008, Dolní Maxov Ing. Petr Koňas, Ph.D.

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